A convex body in \(\mathbb {R}^d\)  is called lattice-free if it does not contain any integer points in its interior.
Lattice-free convex bodies appear in many important works concerning the theory of integer programming.
They are central objects in cutting plane theory [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]} and play a crucial role in Lenstra's algorithm [8]} for integer programming in fixed dimension.
A fundamental property of lattice-free convex bodies is that they are “flat” with respect to the integer lattice.
The lattice width of a convex body \(K \subseteq \mathbb {R}^d\)  is
\(\operatorname{lw}(K) := \min \left\lbrace  \max _{x \in K} \langle x,y \rangle - \min _{x \in K} \langle x,y \rangle : y \in \mathbb {Z}^d \setminus \lbrace  \mathbf {0}\rbrace  \right\rbrace ,\) 
